We therefore take a page out of the one-dimensional power estimation playbook of Thomson. ( ) by forming the âeigenvalue-weighted multitaper estimate.
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis Frederik J. Simons Department of Geosciences, Princeton University, Princeton, NJ, USA
Introduction. . . . .. . . . .. . . .. . . .. . . . .. . . .. . . .. . . . .. . . .. . . . .. . . .. . . .. . . . .. . . .. . . .. . . . .. . .
. .. .. . .. .. . .. .. .
Theory of Slepian Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatiospectral Concentration for Time Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Theory in One Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sturm–Liouville Character and Tridiagonal Matrix Formulation. . . . . . . . . . . . . . . . . . . . Spatiospectral Concentration in the Cartesian Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Theory in Two Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sturm–Liouville Character and Tridiagonal Matrix Formulation. . . . . . . . . . . . . . . . . . . . Spatiospectral Concentration on the Surface of a Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Theory in “Three” Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sturm–Liouville Character and Tridiagonal Matrix Formulation. . . . . . . . . . . . . . . . . . . . Mid-Term Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.. .. .. .. .. ..
Problems in the Geosciences and Beyond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem (i): Signal Estimation from Noisy and Incomplete Spherical Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Harmonic Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slepian Basis Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bias and Variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem (ii): Power Spectrum Estimation from Noisy and Incomplete Spherical Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Spherical Periodogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bias of the Periodogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variance of the Periodogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Spherical Multitaper Estimate.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bias of the Multitaper Estimate.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variance of the Multitaper Estimate.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. ..
Practical Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem (i), Revisited.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Harmonic Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.. .. .. .
W. Freeden, M.Z. Nashed, T. Sonar (Eds.), Handbook of Geomathematics, DOI ./----_, © Springer-Verlag Berlin Heidelberg
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
.. .. . .. ..
Slepian Basis Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bias and Variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem (ii), Revisited.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Spherical Periodogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Spherical Multitaper Estimate.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions. . . . .. . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . ..
References. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
Abstract It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences, we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing data of this kind will be facilitated if a basis of functions can be found that are “spatiospectrally” concentrated, i.e., “localized” in both domains at the same time. Here, we give a theoretical overview of one particular approach to this “concentration” problem, as originally proposed for time series by Slepian and coworkers, in the s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere. Key words: Inverse theory, Satellite geodesy, Sparsity, Spectral analysis, Spherical harmonics, Statistical methods
Introduction
It is well atppreciated that functions cannot have finite support in the temporal (or spatial) and spectral domains at the same time (Slepian ). Finding and representing signals that are optimally concentrated in both is a fundamental problem in information theory which was solved in the early s by Slepian, Landau, and Pollak (Slepian and Pollak ; Landau and Pollak , ). The extensions and generalizations of this problem (Daubechies , ; Daubechies and Paul ; Cohen ) have strong connections with the burgeoning field of wavelet analysis. In this contribution, however, we shall not talk about wavelets, the scaled translates of a “mother” with vanishing moments, the tool for multi-resolution analysis (Daubechies ; Flandrin ; Mallat ). Rather, we devote our attention entirely to what we shall collectively refer to as “Slepian functions,” in multiple Cartesian dimensions and on the sphere. These we understand to be orthogonal families of functions that are all defined on a common, e.g., geographical, domain, where they are either optimally concentrated or within which they are exactly limited, and which at the same time are exactly confined within a certain bandwidth, or maximally concentrated therein. The measure of concentration is invariably a quadratic energy ratio, which, though only one choice out of many (Donoho and Stark ; Riedel and Sidorenko ; Freeden and Windheuser ), is perfectly suited to the nature of the problems we are attempting to address. These are, for example: How do we make estimates of signals that are noisily and incompletely observed? How do we analyze the properties of such signals efficiently, and how can we represent them economically? How do we estimate the power spectrum of noisy and incomplete data? What are the particular constraints imposed by dealing with potential-field signals (gravity, magnetism, etc.) and how is the altitude of the observation point, e.g., from a satellite in orbit, taken into account? What are the statistical properties of the resulting signal and power spectral estimates? These and other questions have been studied extensively in one dimension, that is, for time series, but remarkably little work had been done in the Cartesian plane or on the surface of the sphere. For the geosciences, the latter two domains of application are nevertheless vital for the
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
obvious reasons that they deal with information (measurement and modeling) that is geographically distributed on (a portion of) a planetary surface. In our own series of papers (Wieczorek and Simons ; Simons et al. ; Simons and Dahlen ) we have dealt extensively with Slepian’s problem in spherical geometry; asymptotic reductions to the plane (Simons et al. ; Simons and Dahlen ) then generalize Slepian’s early treatment of the multidimensional Cartesian case (Slepian ). In this chapter, we provide a framework for the analysis and representation of geoscientific data by means of Slepian functions defined for time series, on two-dimensional Cartesian, and spherical domains. We emphasize the common ground underlying the construction of all Slepian functions, discuss practical algorithms, and review the major findings of our own recent work on signal (Wieczorek and Simons ; Simons and Dahlen ) and power spectral estimation theory on the sphere (Wieczorek and Simons ; Dahlen and Simons ).
Theory of Slepian Functions
In this section, we review the theory of Slepian functions in one dimension, in the Cartesian plane, and on the surface of the unit sphere. The one-dimensional theory is quite well known and perhaps most accessibly presented in the textbook by Percival and Walden (). It is briefly reformulated here for consistency and to establish some notation. The two-dimensional planar case formed the subject of a lesser-knowln of Slepian’s papers (Slepian ) and is reviewed here also. We are not discussing alternatives by which two-dimensional Slepian functions are constructed by forming the outer product of pairs of one-dimensional functions. While this approach has produced some useful results (Hanssen ; Simons et al. ), it does not solve the concentration problem sensu stricto. The spherical case was treated in most detail, and for the first time, by ourselves elsewhere (Wieczorek and Simons ; Simons et al. ; Simons and Dahlen ), though two very important early studies by Slepian, Grünbaum, and others, laid much of the foundation for the analytical treatment of the spherical concentration problem for cases with special symmetries (Gilbert and Slepian ; Grünbaum ). Finally, we recast the theory in the context of reproducing-kernel Hilbert spaces, through which the reader may appreciate some of the connections with radial basis functions, splines, and wavelet analysis, which are commonly formulated in such a framework (Freeden et al. ), see also > Chap. of this book.
.
Spatiospectral Concentration for Time Series
..
General Theory in One Dimension
We use t to denote time or one-dimensional space and ω for angular frequency, and adopt a normalization convention (Mallat ) in which a real-valued time-domain signal f (t) and its Fourier transform F(ω) are related by f (t) = (π)− ∫
∞
−∞
F(ω)e i ωt dω,
F(ω) = ∫
∞
−∞
f (t)e−i ωt dt.
()
The problem of finding the strictly bandlimited signal g(t) = (π)− ∫
W
−W
G(ω)e i ωt dω,
()
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
that is maximally, though by virtue of the Paley–Wiener theorem (Daubechies ; Mallat ), never completely concentrated into a time interval ∣t∣ ≤ T was first considered by Slepian, Landau, and Pollak (Slepian and Pollak ; Landau and Pollak ). The optimally concentrated signal is taken to be the one with the least energy outside of the interval: λ=
T
∫ ∫
−T ∞
−∞
g (t) dt
= maximum.
g (t) dt
()
Bandlimited functions g(t) satisfying the variational problem () have spectra G(ω) that satisfy the frequency-domain convolutional integral eigenvalue equation ∫
W
−W
D(ω, ω′) G(ω′ ) dω′ = λG(ω), D(ω, ω′) =
∣ω∣ ≤ W,
sin T(ω − ω′ ) . π(ω − ω′ )
(a) (b)
The corresponding time- or spatial-domain formulation is ∫
T
D(t, t ′) g(t ′ ) dt ′ = λ g(t),
−T
t ∈ ℝ,
sin W(t − t ′) . π(t − t ′)
D(t, t ′) =
(a) (b)
The “prolate spheroidal eigentapers” g (t), g (t), . . . that solve Eq. form a doubly orthogonal set. When they are chosen to be orthonormal over infinite time ∣t∣ ≤ ∞ they are also orthogonal over the finite interval ∣t∣ ≤ T: ∫
∞
−∞
g α g β dt = δ α β ,
∫
T
−T
g α g β dt = λ αh δ α β .
()
A change of variables and a scaling of the eigenfunctions transforms Eq. into the dimensionless eigenproblem ∫
−
′
′
′
D(x, x ) ψ(x ) dx = λψ(x),
D(x, x ′) =
(a)
′
sin T W(x − x ) . π(x − x ′ )
(b)
Equation shows that the eigenvalues λ > λ > ⋯ and suitably scaled eigenfunctions ψ (x), ψ (x), . . . depend only upon the time-bandwidth product T W. The sum of the concentration eigenvalues λ relates to this product by N
D
∞
α=
−
= ∑ λα = ∫
D(x, x) dx =
(T)(W) T W = . π π
()
The eigenvalue spectrum of Eq. has a characteristic step shape, showing significant (λ ≈ ) and insignificant (λ ≈ ) eigenvalues separated by a narrow transition band (Landau ; Slepian and Sonnenblick ). Thus, this “Shannon number” is a good estimate of the number of significant eigenvalues, or, roughly speaking, N D is the number of signals f (t) that can be simultaneously well concentrated into a finite time interval ∣t∣ ≤ T and a finite frequency interval ∣ω∣ ≤ W. In other words (Landau and Pollak ), N D is the approximate dimension of the space of signals that is “essentially” timelimited to T and bandlimited to W, and using the orthogonal set g , g , . . . , g N D as its basis is parsimonious.
..
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
Sturm–Liouville Character and Tridiagonal Matrix Formulation
The integral operator acting upon ψ in Eq. commutes with a differential operator that arises in expressing the three-dimensional scalar wave equation in prolate spheroidal coordinates (Slepian and Pollak ; Slepian ), which makes it possible to find the scaled eigenfunctions ψ by solving the Sturm–Liouville equation (N D ) π d dψ [( − x ) ] + [χ − x ] ψ = , dx dx
()
where χ ≠ λ is the associated eigenvalue. The eigenfunctions ψ(x) of Eq. can be found at discrete values of x by diagonalization of a simple symmetric tridiagonal matrix (Slepian ; Grünbaum ; Percival and Walden ) with elements
([N − − x]/) cos(πW) for x(N − x)/ for
x = , . . . , N − , x = , . . . , N − .
()
The matching eigenvalues λ can then be obtained directly from Eq. . The discovery of the Sturm–Liouville formulation of the concentration problem posed in Eq. proved to be a major impetus for the widespread adoption and practical applications of the “Slepian” basis in signal identification, spectral and numerical analysis. Compared to the sequence of eigenvalues λ, the spectrum of the eigenvalues χ is extremely regular and thus the solution of Eq. is without any problem amenable to finite-precision numerical computation (Percival and Walden ).
.
Spatiospectral Concentration in the Cartesian Plane
..
General Theory in Two Dimensions
A square-integrable function f (x) defined in the plane has the two-dimensional Fourier representation f (x) = (π)− ∫
∞
−∞
F(k)e ik⋅x dk,
F(k) = ∫
∞
−∞
f (x)e −ik⋅x dx,
()
We use g(x) to denote a function that is bandlimited to K, an arbitrary subregion of spectral space, g(x) = (π)−∫ G(k)e ik⋅x dk. () K
Following Slepian (), we seek to concentrate the power of g(x) into a finite spatial region R ∈ ℝ , of area A: ∫ g (x) dx R = maximum. () λ= ∞ g (x) dx ∫ −∞
Bandlimited functions g(x) that maximize the Rayleigh quotient () solve the Fredholm integral equation ′ ′ ′ ∫ D(k, k ) G(k ) dk = λ G(k), K
′
−
D(k, k ) = (π)
∫ e R
i(k′ −k)⋅x
k ∈ K,
(a)
dx.
(b)
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
The corresponding problem in the spatial domain is ′ ′ ′ ∫ D(x, x ) g(x ) dx = λ g(x),
x ∈ ℝ ,
R
(a)
′
D(x, x′) = (π)− ∫ e ik⋅(x−x ) dk.
(b)
K
The bandlimited spatial-domain eigenfunctions g (x), g (x), . . . and eigenvalues λ ≥ λ ≥ . . . that solve Eq. may be chosen to be orthonormal over the whole plane ∥x∥ ≤ ∞ in which case they are also orthogonal over R: ∫
∞
−∞
g α g β dx = δ α β ,
∫
R
g α g β dx = λ α δ α β .
()
Concentration to the disk-shaped spectral band K = {k : ∥k∥ ≤ K} allows us to rewrite Eq. after a change of variables and a scaling of the eigenfunctions as ∫
D(ξ, ξ′ ) ψ(ξ′ ) d ξ′ = λ ψ(ξ),
(a)
R∗
D(ξ, ξ′ ) =
K
√
A/π J (K π
√
′
A/π ∥ξ − ξ ∥) , ∥ξ − ξ′ ∥
(b)
where the region R∗ is scaled to area π and J is the first-order Bessel function of the first kind. Equation () shows that, also in the two-dimensional case, the eigenvalues λ , λ , . . . and the scaled eigenfunctions ψ (ξ), ψ (ξ), . . . depend only on the combination of the circular bandwidth K and the spatial concentration area A, where the quantity K A/(π) now plays the role of the time-bandwidth product T W in the one-dimensional case. The sum of the concentration eigenvalues λ defines the two-dimensional Shannon number N D as N
D
∞
= ∑ λ α = ∫ D(ξ, ξ) d ξ = α=
(πK )(A)
R∗
(π)
=K
A . π
()
Just as N D in Eq. , N D is the product of the spectral and spatial areas of concentration multiplied by the “Nyquist density” (Daubechies , ). And, similarly, it is the effective dimension of the space of “essentially” space- and bandlimited functions in which the set of two-dimensional functions g , g , . . . , g N D may act as a sparse orthogonal basis. An example of Slepian functions on a geographical domain in the Cartesian plane can be found in > Fig. .
..
Sturm–Liouville Character and Tridiagonal Matrix Formulation
If in addition to the circular spectral limitation, space is also circularly limited, in other words, if the spatial region of concentration or limitation R is a circle of radius R, then a polar coordinate, x = (r, θ), representation √ ⎧ ⎪ g(r) cos mθ ⎪ ⎪ ⎪ ⎪ ⎪ g(r, θ) = ⎨ g(r) ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎩ g(r) sin mθ
if m < , if m = , if m > ,
()
− 500
Northing (km)
0
500
kx / k Nyq x
0
Easting (km)
0.1
− 0.1
0
500
kx / k Nyq x
0
Easting (km)
− 500
l 2 = 0.957996
0.1
− 0.1
− 500
500
kx / k Nyq x
0
Easting (km)
0
l 3 = 0.936969
0.1
− 0.1
− 500
500
kx / k Nyq x
0
Easting (km)
0
l 4 = 0.894682
0.1
⊡ Fig. Bandlimited eigenfunctions g , g , . . . , g that are optimally concentrated within the Columbia Plateau, a physiographic region in the United States centered on .○ W .○ N (near Boise City, Idaho) of area A ≈ × km . The concentration factors λ , λ , . . . , λ are indicated; the Shannon number ND = . The top row shows a rendition of the eigenfunctions in space on a grid with km resolution in both directions, with the convention that positive values are blue and negative values red, though the sign of the functions is arbitrary. The spatial concentration region is outlined in black. The bottom row shows the squared Fourier coefficients ∣Gα (k)∣ as calculated from the functions gα (x) shown, on a wavenumber scale that is expressed as a fraction of the Nyquist wavenumber. The spectral limitation region is shown by the black circle at wavenumber K = . rad/km. All areas for which the absolute value of the functions plotted is less than one hundredth of the maximum value attained over the domain are left white. The calculations were performed by the Nyström method using Gauss–Legendre integration of Eq. in the two-dimensional spatial domain
− 0.1 − 0.1
0
0.1
− 400
0
400
l 1 = 0.971268
k y / k yNyq
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
may be used to decompose Eq. into a series of non-degenerate fixed-order eigenvalue problems, after scaling, ∫
′
′
′
′
D(ξ, ξ )ψ(ξ ) ξ d ξ = λ ψ(ξ),
(a)
√ √ D(ξ, ξ ′) = N ∫ J m ( N D p ξ) J m ( N D p ξ ′ ) p d p.
(b)
Solutions to Eq. solve a Sturm–Liouville equation on ≤ ξ ≤ . In terms of φ(ξ) = dφ / − m d − N D ξ ) φ = , [( − ξ ) ] + (χ + dξ dξ ξ
√
ξ ψ(ξ), ()
for some χ ≠ λ. When m = ±/, Eq. reduces to Eq. . By extension to ξ > the fixed-order “generalized prolate spheroidal functions” φ (ξ), φ (ξ), . . . can be determined from the rapidly converging infinite series φ(ξ) =
m! ∞ d l l! J m+l + (c ξ) , √ ∑ γ l = (l + m)! cξ
ξ ∈ ℝ,
()
where the fixed-m expansion coefficients d l are determined by recursion (Slepian ) or by diagonalization of a non-symmetric tridiagonal matrix (de Villiers et al. ; Shkolnisky ) with elements given by Tl +l = −
c (m + l + ) , (l + m + )(l + m + )
c m ], Tl l = (l + m + ) (l + m + ) + [ + (l + m)(l + m + ) Tl l + = −
c (l + ) , (l + m + )(l + m + )
()
where the parameter l ranges from to some large value that ensures convergence. The desired concentration eigenvalues λ can subsequently be obtained by direct integration of Eq. , or, alternatively, from
λl = γ
√
−
N D ,
with
γ=
∞ c m+/ d dl ) . ( ∑ m+ (m + )! l =
()
Numerous numerical methods exist to use Eqs. – and in solving the concentration problem () for a variety of two-dimensional Cartesian domains. After a long hiatus since the work of Slepian (), the two-dimensional problem has recently been the focus of renewed attention in the applied mathematics community (de Villiers et al. ; Shkolnisky ), and applications to the geosciences are to follow (Simons and Wang ). An example of Slepian functions on a disk-shaped region in the Cartesian plane can be found in > Fig. .
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
l 1 = 1.000000
l 2 = 1.000000
l 6 = 0.999997
l 7 = 0.999974
l
= 0.999738
l
l
= 0.998029
l
11
16
12
17
l 3 = 1.000000
l 8 = 0.999974
= 0.999738
l
= 0.998028
l
13
18
l 4 = 0.999998
l 5 = 0.999998
l 9 = 0.999930
l 10 = 0.999930
= 0.999121
l
= 0.992470
l
14
19
= 0.999121
l
= 0.992470
l
15
20
= 0.998757
= 0.988706
l 21 = 0.988701
l 22 = 0.986945
l 23 = 0.986930
l 24 = 0.955298
l25 = 0.955287
l
l
l
l
l
26
= 0.951126
27
= 0.951109
28
= 0.915710
29
= 0.915709
30
= 0.898353
⊡ Fig. Bandlimited eigenfunctions gα (r, θ) that are optimally concentrated within a Cartesian disk of radius R = . The dashed circle denotes the region boundary. The Shannon number ND = . The eigenvalues λ α have been sorted to a global ranking with the best concentrated eigenfunction plotted at the top left and the th best in the lower right. Blue is positive and red is negative and the color axis is symmetric, but the sign is arbitrary; regions in which the absolute value is less than one hundredth of the maximum value on the domain are left white. The calculations were performed by Gauss–Legendre integration in the two-dimensional spatial domain, which sometimes leads to slight differences in the last two digits of what should be identical eigenvalues for each pair of non-circularly-symmetric eigenfunctions
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
.
Spatiospectral Concentration on the Surface of a Sphere
..
General Theory in “Three” Dimensions
We denote the colatitude of a geographical point ˆr on the unit sphere Ω = {ˆr : ∥ˆr∥ = } by ≤ θ ≤ π and the longitude by ≤ ϕ < π. We use R to denote a region of Ω, of area A, within which we seek to concentrate a bandlimited function of position ˆr. We express a square-integrable function f (ˆr) on the surface of the unit sphere as ∞
l
f (ˆr) = ∑ ∑ f l m Yl m (ˆr),
f l m = ∫ f (ˆr)Yl m (ˆr) dΩ, Ω
l = m=−l
using real surface spherical harmonics (Edmonds ; Dahlen and Tromp ) √ ⎧ ⎪ X l m (θ) cos mϕ if −l ≤ m < , ⎪ ⎪ ⎪ ⎪ ⎪ Yl m (ˆr) = Yl m (θ, ϕ) = ⎨ X l (θ) if m = , ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ X l m (θ) sin mϕ if < m ≤ l, X l m (θ) = (−)m ( Pl m (μ) =
l + / (l − m)! ) [ ] π (l + m)!
d ( − μ )m/ ( ) l dμ l!
()
()
/
Pl m (cos θ),
()
l +m
(μ − ) l .
()
The quantity ≤ l ≤ ∞ is the angular degree of the spherical harmonic, and −l ≤ m ≤ l is its angular order. The function Pl m (μ) defined in () is the associated Legendre function of integer degree l and order m. Our choice of the constants in Eqs. – orthonormalizes the harmonics on the unit sphere: ∫ Yl m Yl ′ m′ dΩ = δ l l ′ δ mm′ , Ω
()
and leads to the addition theorem in terms of the Legendre functions Pl (μ) = Pl (μ) as l
′ ∑ Yl m (ˆr)Yl m (ˆr ) = ( m=−l
l + ) Pl (ˆr ⋅ ˆr′ ). π
()
To maximize the spatial concentration of a bandlimited function L
l
g(ˆr) = ∑ ∑ g l m Yl m (ˆr)
()
l = m=−l
within a region R, we maximize the energy ratio
λ=
∫ g (ˆr) dΩ R
∫ g (ˆr) dΩ
= maximum.
()
Ω
Maximizing Eq. leads to the positive-definite spectral-domain eigenvalue equation L
l′
∑ ∑ D l m,l ′ m′ g l ′ m′ = λ g l m ,
l ′ = m′ =−l ′
≤ l ≤ L,
D l m,l ′ m′ = ∫ Yl m Yl ′ m′ dΩ, R
(a) (b)
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
and we may equally well rewrite Eq. as a spatial-domain eigenvalue equation: ′ ′ ′ ∫ D(ˆr, ˆr ) g(ˆr ) dΩ = λ g(ˆr),
ˆr ∈ Ω,
R
L
D(ˆr, ˆr′ ) = ∑ ( l =
l + )Pl (ˆr ⋅ ˆr′ ), π
(a) (b)
where Pl is the Legendre function of degree l. Equation () is a homogeneous Fredholm integral equation of the second kind, with a finite-rank, symmetric, Hermitian kernel. We choose the eigenfunctions of the operator in Eq. b, whose elements are g l m α , α = , . . . , (L + ) , to satisfy the orthonormality relations ∑ gl m α gl m β = δα β,
∑ g l m α ∑ D l m,l ′ m′ g l ′ m′ β = λ α δ α β .
L
L
lm
lm
l ′ m′
L
()
The finite set of bandlimited spatial eigensolutions g (ˆr), g (ˆr), . . . , g(L+) (ˆr) can be made orthonormal over the whole sphere Ω and orthogonal over the region R: ∫ g α g β dΩ = δ α β ,
∫ g α g β dΩ = λ α δ α β .
Ω
R
()
In the limit of a small area A → and a large bandwidth L → ∞ and after a change of variables, a scaled version of Eq. will be given by ′
′
′
∫ D(ξ, ξ ) ψ(ξ ) dΩ∗ = λ ψ(ξ),
(a)
√ √ (L + ) A/π J [(L + ) A/π ∥ξ − ξ′ ∥] , π ∥ξ − ξ′ ∥
(b)
R∗
D(ξ, ξ′ ) =
where the scaled region R∗ now has area π and J again is the first-order Bessel function of the first kind. As in the one- and two-dimensional case, the asymptotic, or “flat-Earth” eigenvalues λ ≥ λ ≥ ⋯ and scaled eigenfunctions ψ (ξ), ψ (ξ), . . . depend upon the maximal degree L and the area A only through what is once again a space-bandwidth product, the “spherical Shannon number”, this time given by (L+)
L
l
N D = ∑ λ α = ∑ ∑ D l m,l m = ∫ D(ˆr, ˆr) dΩ α=
l = m=−l
R
A = ∫ D(ξ, ξ) dΩ∗ = (L + ) . π R∗
()
Irrespectively of the particular region of concentration that they were designed for, the complete set of bandlimited spatial Slepian eigenfunctions g , g , . . . , g(L+) , is a basis for bandlimited scalar processes anywhere on the surface of the unit sphere (Simons et al. ; Simons and Dahlen ). This follows directly from the fact that the spectral localization kernel (b) is real, symmetric, and positive definite: its eigenvectors g l m , g l m , . . . , g(L+) l m form an orthogonal set as we have seen. Thus the Slepian basis functions g α (ˆr), α = , . . . , (L + ) given by Eq. simply transform the same-sized limited set of spherical harmonics Yl m (ˆr), ≤ l ≤ L, −l ≤ m ≤ l that are a basis for the same space of bandlimited spherical functions with no power above the bandwidth L. The effect of this transformation is to order the resulting basis set such that the energy of the first N D functions, g (ˆr), . . . , g N D (ˆr), with eigenvalues λ ≈ , is concentrated in the region R, whereas the remaining eigenfunctions, g N D + (ˆr), . . . , g(L+) (ˆr), are
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
concentrated in the complimentary region Ω − R. As in the one- and two-dimensional case, therefore, the reduced set of basis functions g , g , . . . , g N D can be regarded as a sparse, global, basis suitable to approximate bandlimited processes that are primarily localized to the region R. The dimensionality reduction is dependent on the fractional area of the region of interest. In other words, the full dimension of the space (L+) can be “sparsified” to an effective dimension of N D = (L + ) A/(π) when the signal resides in a particular geographic region. An example of Slepian functions on a geographical domain on the surface of the sphere is found in > Fig. .
−65° −75° −85° −95° −105°
l = 1.000
l = 1.000
l = 1.000
l = 1.000
l = 1.000
l = 1.000
l = 1.000
l = 1.000
l = 1.000
l = 1.000
l = 1.000
l = 1.000
−115° −65° −75° −85° −95° −105° −115° −65° −75° −85° −95° −105° −115° −65° −75° −85° −95° −105°
−115° −30° −15°
0°
15°
30°
−30° −15°
0°
15°
30°
−30° −15°
0°
15°
30°
⊡ Fig. Bandlimited L = eigenfunctions g , g , . . . , g that are optimally concentrated within Antarctica. The concentration factors λ , λ , . . . , λ are indicated; the rounded Shannon number is ND = . The order of concentration is left to right, top to bottom. Positive values are blue and negative values are red; the sign of an eigenfunction is arbitrary. Regions in which the absolute value is less than one hundredth of the maximum value on the sphere are left white. We integrated Eq. b over a splined high-resolution representation of the boundary, using Gauss–Legendre quadrature over the colatitudes, and analytically in the longitudinal dimension (Simons and Dahlen )
..
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
Sturm–Liouville Character and Tridiagonal Matrix Formulation
In the special but important case in which the region of concentration is a circularly symmetric cap of colatitudinal radius Θ, centered on the North Pole, the colatitudinal parts g(θ) of the separable functions √ ⎧ ⎪ g(θ) cos mϕ if −L ≤ m < , ⎪ ⎪ ⎪ ⎪ ⎪ g(θ, ϕ) = ⎨ g(θ) () if m = , ⎪ ⎪ √ ⎪ ⎪ ⎪ g(θ) sin mϕ if < m ≤ L, ⎪ ⎩ which solve Eq. , or, indeed, the fixed-order versions ∫
Θ
D(θ, θ ′) g(θ ′ ) sin θ ′ dθ ′ = λ g(θ),
≤ θ ≤ π,
(a)
L
D(θ, θ ′) = π ∑ X l m (θ)X l m (θ ′ ),
(b)
l =m
are identical to those of a Sturm–Liouville equation for the g(θ). In terms of μ = cos θ, dg m (μ − cos Θ) d ) g = , [(μ − cos Θ)( − μ ) ] + (χ + L(L + )μ − dμ dμ − μ
()
with χ ≠ λ. This equation can be solved in the spectral domain by diagonalization of a simple symmetric tridiagonal matrix with a very well-behaved spectrum (Simons et al. ; Simons and Dahlen ). This matrix, whose eigenfunctions correspond to the g l m of Eq. at constant m, is given by Tl l = −l(l + ) cos Θ, Tl l +
" # (l + ) − m $ = [l(l + ) − L(L + )]# . (l + )(l + )
()
Moreover, when the region of concentration is a pair of axisymmetric polar caps of common colatitudinal radius Θ centered on the North and South Pole, the g(θ) can be obtained by solving the Sturm–Liouville equation dg m (μ − cos Θ) d ) g = , [(μ − cos Θ)( − μ ) ] + (χ + L p (L p + )μ − dμ dμ − μ
()
where L p = L or L p = L − depending whether the order m of the functions g(θ, ϕ) in Eq. is odd or even, and whether the bandwidth L itself is odd or even. In their spectral form the coefficients of the optimally concentrated antipodal polar-cap eigenfunctions only require the numerical diagonalization of a symmetric tridiagonal matrix with analytically prescribed elements and a spectrum of eigenvalues that is guaranteed to be simple (Simons and Dahlen , ), namely p [(l + ) − m ] Tl l = −l(l + ) cos Θ + l + m − l(l + ) + [(l − )(l + ) − L p (L p + )] [ − ], (l + )(l − ) " [l(l + ) − L p (L p + )] # # p $ [(l + ) − m ] [(l + ) − m ] Tl l + = . () l + (l + )(l + )
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
The concentration values λ, in turn, can be determined from the defining Eqs. or . The spectra of the eigenvalues χ of Eqs. and display roughly equant spacing, without the numerically troublesome plateaus of nearly equal values that characterizes the eigenvalues λ. Thus, for the special cases of symmetric single and double polar caps, the concentration problem posed in Eq. is not only numerically feasible also in circumstances where direct solution methods are bound to fail (Albertella et al. ), but essentially trivial in every situation. Numerical methods for the solution of Eqs. (–) on completely general domains on the surface of the sphere were discussed by us elsewhere (Simons et al. ; Simons and Dahlen , ).
.
Mid-Term Summary
It is interesting to reflect, however heuristically, on the commonality of all of the above aspects of spatiospectral localization, in the slightly expanded context of reproducing-kernel Hilbert spaces (Yao ; Nashed and Walter ; Daubechies ; Amirbekyan et al. ). In one dimension, the Fourier orthonormality relation and the “reproducing” properties of the spatial delta function are given by ′
−
δ(t, t ) = (π) ∫
∞
−∞
e
i ω(t−t ′ )
dω,
∫
∞
′
′
′
f (t ) δ(t, t ) dt = f (t).
−∞
()
In two Cartesian dimensions, the equivalent relations are ′
−
δ(x, x ) = (π)
∫
∞
−∞
e
ik⋅(x−x′ )
dk,
∫
∞
−∞
′
′
′
f (x ) δ(x, x ) dx = f (x),
()
and on the surface of the unit sphere we have ∞
δ(ˆr, ˆr′ ) = ∑ ( l =
l + )Pl (ˆr ⋅ rˆ′ ), π
′ ′ ′ ∫ f (ˆr ) δ(ˆr, ˆr ) dΩ = f (ˆr). Ω
()
The integral-equation kernels (b), (b) and (b) are all bandlimited spatial delta functions which are reproducing kernels for bandlimited functions of the types in Eqs. , and : D(t, t ′) = (π)− ∫
W
−W
′
−
D(x, x ) = (π) ′
L
D(ˆr, ˆr ) = ∑ ( l =
∫ e K
′
e i ω(t−t ) dω, ik⋅(x−x′ )
dk,
l + ′ )Pl (ˆr ⋅ ˆr ), π
∞
∫
−∞ ∞
∫
−∞
g(t ′ )D(t, t ′) dt ′ = g(t), ′
′
′
g(x )D(x, x ) dx = g(x), ′
′
∫ g(ˆr )D(ˆr, ˆr ) dΩ = g(ˆr). Ω
() () ()
The equivalence of Eq. with Eq. b is through the Euler identity and the reproducing properties follow from the spectral forms of the orthogonality relations (–), which are self-evident by change-of-variables, and from the spectral form of Eq. , which is Eq. . Much as the delta functions of Eqs. – set up the Hilbert spaces of all square-integrable functions on the real line, in two-dimensional Cartesian space, and on the surface of the sphere, the kernels (–) induce the equivalent subspaces of bandlimited functions in their respective dimensions. Inasmuch as the Slepian functions are the integral eigenfunctions of these reproducing kernels in the sense of Eqs. a, a, and a, they are complete bases for their band-limited subspaces (Slepian and Pollak ; Landau and Pollak ; Daubechies ; Flandrin ; Freeden et al. ).
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
Therein, the N D , N D , or N D best time- or space-concentrated members allow for sparse, approximate, expansions of signals that are spatially concentrated to the one-dimensional interval t ∈ [−T, T] ⊂ ℝ, the Cartesian region x ∈ R ⊂ ℝ , or the spherical surface patch ˆr ∈ R ⊂ Ω. As a corollary to this behavior, the infinite sets of exactly time- or spacelimited (and thus band-concentrated) versions of the functions g, which are the eigenfunctions of Eqs. , and with the domains appropriately restricted, are complete bases for square-integrable scalar functions on the intervals to which they are confined (Slepian and Pollak ; Landau and Pollak ; Simons et al. ). Expansions of such wideband signals in the small subset of their N D , N D , or N D most band-concentrated members provide reconstructions which are constructive in the sense that they progressively capture all of the signal in the mean-squared sense, in the limit of letting their number grow to infinity. This second class of functions can be trivially obtained, up to a multiplicative constant, from the bandlimited Slepian functions g by simple time- or space limitation. While Slepian (Slepian and Pollak ; Slepian ), for this reason perhaps, never gave them a name, we have been referring to those as h in our own investigations of such functions on the sphere (Simons et al. ; Simons and Dahlen ; Dahlen and Simons ).
Problems in the Geosciences and Beyond
Taking all of the above at face-value but referring again to the literature cited thus far for proof and additional context, we return to considerations closer to home, namely the estimation of geophysical (or cosmological) signals and/or their power spectra, from noisy and incomplete observations collected at or above the surface of the spheres “Earth” or “planet” (or from inside the sphere “sky”). We restrict ourselves to real-valued scalar measurements, contaminated by additive noise for which we shall adopt idealized models. We focus exclusively on data acquired and solutions expressed on the unit sphere. We have considered generalizations to problems involving satellite data collected at an altitude and/or potential fields elsewhere (Simons and Dahlen ; Simons et al. ). Two different statistical problems come up in this context, namely, (i) how to find the “best” estimate of the signal given the data, and (ii) how to construct from the data the “best” estimate of the power spectral density of the signal in question. Thus, let there be some data distributed on the unit sphere, consisting of “signal,” s and “noise,” n, and let there be some region of interest R ⊂ Ω, in other words, let d(ˆr) = {
s(ˆr) + n(ˆr) unknown/undesired
if ˆr ∈ R, if ˆr ∈ R − Ω.
()
We assume that the signal of interest can be expressed by way of spherical harmonic expansion as in Eq. , and that it is, itself, a realization of a zero-mean, Gaussian, isotropic, random process, namely ∞
l
s(ˆr) = ∑ ∑ s l m Yl m (ˆr), l = m=−l
⟨s l m ⟩ =
and ⟨s l m s l ′ m′ ⟩ = S l δ l l ′ δ mm′ .
()
For illustration we furthermore assume that the noise is a zero-mean stochastic process with an isotropic power spectrum, i.e., ⟨n(ˆr)⟩ = and ⟨n l m n l ′ m′ ⟩ = N l δ l l ′ δ mm′ , and that it is
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
statistically uncorrelated with the signal. We refer to power as white when S l = S or N l = N, or, equivalently, when ⟨n(ˆr)n(ˆr′ )⟩ = N δ(ˆr, ˆr′ ). Our objectives are thus (i) to determine the best estimate sˆl m of the spherical harmonic expansion coefficients s l m of the signal and (ii) to find the best estimate Sˆl for the isotropic power spectral density S l . While in the physical world there can be no limit on bandwidth, practical restrictions force any and all of our estimates to be bandlimited to some maximum spherical harmonic degree L, thus by necessity sˆl m = and Sˆl = for l > L: L
l
sˆ(ˆr) = ∑ ∑ sˆl m Yl m (ˆr).
()
l = m=−l
This limitation, combined with the statements Eq. on the data coverage or the study region of interest, naturally leads us back to the concept of “spatiospectral concentration,” and, as we shall see, solving either problem (i) or (ii) will gain from involving the “localized” Slepian functions rather than, or in addition to, the “global” spherical harmonics basis. This leaves us to clarify what we understand by “best” in this context. We adopt the traditional statistical metrics of bias, variance, and mean-squared error to appraise the quality of our solutions (Cox and Hinkley ; Bendat and Piersol ), in anticipation that the Slepian functions will be naturally instrumental in constructing efficient, consistent and/or unbiased estimates of either sˆl m or Sˆl . Thus, we define
v = ⟨ˆs ⟩ − ⟨ˆs⟩ ,
b = ⟨ˆs⟩ − s,
є = sˆ − s,
⟨є ⟩ = v + b
and
()
for problem (i), where the lack of subscripts indicates that we can study variance, bias and meansquared error of the estimate of the coefficients sˆl m but also of their spatial expansion sˆ(ˆr). For problem (ii) on the other hand, we focus on the estimate of the isotropic power spectrum at a given spherical harmonic degree l by identifying v l = ⟨ Sˆl ⟩ − ⟨Sˆl ⟩ ,
b l = ⟨ Sˆl ⟩ − S l ,
є l = Sˆl − S l ,
and ⟨є l ⟩ = v l + b l .
()
Depending on the application, the “best” estimate could mean the unbiased one with the lowest variance (Tegmark ; Tegmark et al. ; Bond et al. ; Oh et al. ; Hinshaw et al. ), it could be simply the minimum-variance estimate having some acceptable and quantifiable bias (Wieczorek and Simons ), or, as we would usually prefer, it would be the one with the minimum mean-squared error (Simons and Dahlen ; Dahlen and Simons ).
.
Problem (i): Signal Estimation from Noisy and Incomplete Spherical Data
..
Spherical Harmonic Solution
Paraphrasing results elaborated elsewhere (Simons and Dahlen ), we write the bandlimited solution to the damped inverse problem ∫ (s − d) dΩ + η ∫ ¯ s dΩ = minimum, R
R
()
where η ≥ is a damping parameter, by straightforward algebraic manipulation, as L
l′
¯ l m,l ′ m′ )− ∫ d Yl ′ m′ dΩ, sˆl m = ∑ ∑ (D l m,l ′ m′ + η D l ′ = m′ =−l ′
R
()
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
¯ l m,l ′ m′ , the kernel that localizes to the region R¯ = Ω − R, compliments D l m,l ′ m′ given where D by Eq. b which localizes to R. Given the eigenvalue spectrum of the latter, its inversion is inherently unstable, thus Eq. is an ill-conditioned inverse problem unless η > , as has been well known, e.g., in geodesy (Xu ; Sneeuw and van Gelderen ). Elsewhere (Simons and Dahlen ) we have derived exact expressions for the optimal value of the damping parameter η as a function of the signal-to-noise ratio under certain simplifying assumptions. As can be easily shown, without damping the estimate is unbiased but effectively incomputable; the introduction of the damping term stabilizes the solution at the cost of added bias. And of course when R = Ω, Eq. is simply the spherical harmonic transform, as in that case, Eq. b reduces to Eq. , in other words, then D l m,l ′ m′ = δ l l ′ δ mm′ .
..
Slepian Basis Solution
The trial solution in the Slepian basis designed for this region of interest R, i.e., (L+)
sˆ(ˆr) = ∑ sˆα g α (ˆr),
()
α=
would be completely equivalent to the expression in Eq. by virtue of the completeness of the Slepian basis for bandlimited functions everywhere on the sphere and the unitarity of the transform () from the spherical harmonic to the Slepian basis. The solution to the undamped (η = ) version of Eq. would then be sˆα = λ− α ∫ d g α dΩ, R
()
which, being completely equivalent to Eq. for η = , would be computable, and biased, only when the expansion in Eq. were to be truncated to some finite J < (L + ) to prevent the blowup of the eigenvalues λ. Assuming for simplicity of the argument that J = N D , the essence of the approach is now that the solution N D
sˆ(ˆr) = ∑ sˆα g α (ˆr)
()
α=
will be sparse (in achieving a bandwidth L using N D Slepian instead of (L + ) sphericalharmonic expansion coefficients) yet good (in approximating the signal as well as possible in the mean-squared sense in the region of interest R) and of geophysical utility, assuming we are dealing with spatially localized processes that are to be extracted, e.g., from global satellite measurements (Han and Simons ; Simons et al. ).
..
Bias and Variance
In concluding this section, let us illustrate another welcome by-product of our methodology, by writing the mean-squared error for the spherical harmonic solution () compared to the equivalent expression for the solution in the Slepian basis, Eq. . We do this as a function of the spatial coordinate, in the Slepian basis for both, and, for maximum clarity of the exposition,
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
using the contrived case when both signal and noise should be white as well as bandlimited (which technically is impossible). In the former case, we get (L+)
−
⟨є (ˆr)⟩ = N ∑ λ α [λ α + η( − λ α )] g α (ˆr)
()
α=
(L+)
−
+ η S ∑ ( − λ α ) [λ α + η( − λ α )] g α (ˆr), α=
while in the latter, we obtain N D
(L+)
α=
α>N D
r) + S ∑ g α (ˆr). ⟨є (ˆr)⟩ = N ∑ λ− α g α (ˆ
()
All (L + ) basis functions are required to express the mean-squared estimation error, whether in Eq. or in Eq. . The first term in both expressions is the variance, which depends on the measurement noise. Without damping or truncation the variance grows without bounds. Damping and truncation alleviate this at the expense of added bias, which depends on the characteristics of the signal, as given by the second term. In contrast to Eq. , however, the Slepian expression () has disentangled the contributions due to noise/variance and signal/bias by projecting them onto the sparse set of well-localized and the remaining set of poorly localized Slepian functions, respectively. The estimation variance is felt via the basis functions α = → N D that are well concentrated inside the measurement area, and the effect of the bias is relegated to those α = N D + → (L + ) functions that are confined to the region of missing data. When forming a solution to problem (i) in the Slepian basis by truncation according to Eq. , changing the truncation level to values lower or higher than the Shannon number N D amounts to navigating the trade-off space between variance, bias (or “resolution”), and sparsity in a manner that is captured with great clarity by Eq. . We refer the reader elsewhere (Simons and Dahlen , ; Simons et al. ) for more details, and, in particular, for the case of potential fields estimated from data collected at satellite altitude.
.
Problem (ii): Power Spectrum Estimation from Noisy and Incomplete Spherical Data
Following Dahlen and Simons () we will find it convenient to regard the data d(ˆr) given in Eq. as having been multiplied by a unit-valued boxcar window function confined to the region R, ∞
p
b(ˆr) = ∑ ∑ b pq Ypq (ˆr) = { p= q=−p
if ˆr ∈ R, otherwise.
()
The power spectrum of the boxcar window () is Bp =
p ∑ b pq . p + q=−p
()
..
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
The Spherical Periodogram
Should we decide that an acceptable estimate of the power spectral density of the available data is nothing else but the weighted average of its spherical harmonic expansion coefficients, we would be forming the spherical analogue of what Schuster () named the “periodogram” in the context of time series analysis, namely l π SP Sˆl = ( ) ∑ [∫ d(ˆr) Yl m (ˆr) dΩ] . A l + m=−l R
..
()
Bias of the Periodogram
Upon doing so we would discover that the expected value of such an estimator would be the biased quantity ∞
⟨SˆSP l ⟩ = ∑ K l l ′ (S l ′ + N l ′ ),
()
l ′ =
where, as it is known in astrophysics and cosmology (Peebles ; Hauser and Peebles ; Hivon et al. ), the periodogram “coupling” matrix ′
K
l l′
l l π =( ) ∑ ∑ [D l m,l ′ m′ ] , A l + m=−l m′ =−l ′
()
governs the extent to which an estimate SˆSP l of S l is influenced by spectral leakage from power in neighboring spherical harmonic degrees l ′ = l ± , l ± , . . ., all the way down to and up to ∞. In the case of full data coverage, R = Ω, or of a perfectly white spectrum, S l = S, however, the estimate would be unbiased, as the rows of the coupling matrix sum to one—provided the noise spectrum, if known, can be subtracted beforehand.
..
Variance of the Periodogram
The covariance of the periodogram estimator () would moreover be suffering from strong wideband coupling of the power spectral densities in being given by Σ SP l l′
⎡∞ ∞ ⎤ l l′ ⎢ ⎥ (π/A) ⎢ = ∑ ∑ ∑ ∑ (S p + N p )D l m, pq D pq,l ′ m′ ⎥ ⎥ . (l + )(l ′ + ) m=−l m′ =−l ′ ⎢ ⎢ p= q= ⎥ ⎣ ⎦
()
Even under the commonly made assumption as should the power spectrum be slowly varying within the main lobe of the coupling matrix, such coupling would be nefarious. In the “locally white” case we would have ′
Σ SP l l′ =
l l (π/A) ′ + Nl′) ∑ + N )(S [D l m,l ′ m′ ] . (S ∑ l l l (l + )(l ′ + ) m=−l m′ =−l ′
()
Only in the limit of whole-sphere data coverage will Eqs. or reduce to WS
Σl l′ =
(S l + N l ) δ l l ′ , l +
()
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
which is the “planetary” or “cosmic” variance that can be understood on the basis of elementary statistical considerations (Jones ; Knox ; Grishchuk and Martin ). The strong spectral leakage for small regions (A ≪ π) is highly undesirable and makes the periodogram “hopelessly obsolete” (Thomson and Chave ), or, to put it kindly, “naive” (Percival and Walden ), just as it is for one-dimensional time series. In principle it is possible—after subtraction of the noise bias—to eliminate the leakage bias in the periodogram estimate () by numerical inversion of the coupling matrix K l l ′ . Such a “deconvolved periodogram” estimator is unbiased. However, its covariance depends on the inverse of the periodogram coupling matrix, which is only feasible when the region R covers most of the sphere, A ≈ π. For any region whose area A is significantly smaller than π, the periodogram coupling matrix () will be too ill-conditioned to be invertible. Thus, much like in problem (i) we are faced with bad bias and poor variance, both of which are controlled by the lack of localization of the spherical harmonics and their non-orthogonality over incomplete subdomains of the unit sphere. Both effects are described by the spatiospectral localization kernel defined in (b), which, in the quadratic estimation problem (ii) appears in “squared” form in Eq. . Undoing the effects of the wideband coupling between degrees at which we seek to estimate the power spectral density by inversion of the coupling kernel is virtually impossible, and even if we could accomplish this to remove the estimation bias, this would much inflate the estimation variance (Dahlen and Simons ).
..
The Spherical Multitaper Estimate
We therefore take a page out of the one-dimensional power estimation playbook of Thomson () by forming the “eigenvalue-weighted multitaper estimate.” We could weight single-taper estimates adaptively to minimize quality measures such as estimation variance or mean-squared error (Thomson ; Wieczorek and Simons ), but in practice, these methods tend to be rather computationally demanding. Instead we simply multiply the data d(ˆr) by the Slepian functions or “tapers” g α (ˆr) designed for the region of interest prior to computing power and then averaging:
(L+) l π MT Sˆl = ∑ λ α ( D ) ∑ [∫ g α (ˆr) d(ˆr) Yl m (ˆr) dΩ] . N l + m=−l Ω α=
..
()
Bias of the Multitaper Estimate
The expected value of the estimate () is l +L
⟨SˆMT l ⟩ = ∑ M l l ′ (S l ′ + N l ′ ),
()
l ′ =l −L
where the eigenvalue-weighted multitaper coupling matrix, using Wigner - j functions (Varshalovich et al. ; Messiah ), is given by Ml l′ =
l ′ + L l ∑ (p + )( (L + ) p=
p
l′ ) .
()
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
It is remarkable that this result depends only upon the chosen bandwidth L and is completely independent of the size, shape, or connectivity of the region R, even as R = Ω. Moreover, every row of the matrix in Eq. sums to unity, which ensures that a the multitaper spectral estimate SˆMT has no leakage bias in the case of a perfectly white spectrum provided the noise bias is l subtracted as welll: ⟨Sˆl MT⟩ − ∑ M l l ′ N l ′ = S if S l = S.
..
Variance of the Multitaper Estimate
Under the moderately colored approximation, which is more easily justified in this case because the coupling () is confined to a narrow band of width less than or equal to L + , with L the bandwidth of the tapers, the eigenvalue-weighted multitaper covariance is Σ MT l l′ =
L l (S l + N l )(S l ′ + N l ′ ) ∑ (p + ) Γp ( π p=
p
l′ ) ,
()
where, using Wigner - j and - j functions (Varshalovich et al. ; Messiah ), Γp =
L
(N D ) L
L
L
L
′ ′ ∑ ∑ ∑ ∑ (s + )(s + )(u + )(u + ) s= s ′ = u= u ′ =
× ∑ (−)
p+e
(e + )B e
e=
×{
s u
e p
s′ s }( u′
e
u s′ )(
e
u′ s )(
p
u′ u )(
p
s′ ) . ()
In this expression B e , the boxcar power (), which we note does depend on the shape of the region of interest, appears again, summed over angular degrees limited by - j selection rules to ≤ e ≤ L. The sum in Eq. is likewise limited to degrees ≤ p ≤ L. The effect of tapering with windows bandlimited to L is to introduce covariance between the estimates at any two different degrees l and l ′ that are separated by fewer than L + degrees. Equation (–) are very efficiently computable, which should make them competitive with, e.g., jackknifed estimates of the estimation variance (Chave et al. ; Thomson and Chave ; Thomson ). The crux of the analysis lies in the fact that the matrix of the spectral covariances between single-tapered estimates is almost diagonal (Wieczorek and Simons ), showing that the individual estimates that enter the weighted formula () are almost uncorrelated statistically. This embodies the very essence of the multitaper method. It dramaticallhy reduces the estimation variance at the cost of small increases of readily quantifiable bias.
Practical Considerations
In this section, we now turn to the very practical context of sampled, e.g., geodetic, data on the sphere. We shall deal exclusively with bandlimited functions, which are equally well expressed in the spherical harmonic as the Slepian basis, namely: L
l
(L+)
f (ˆr) = ∑ ∑ f l m Yl m (ˆr) = ∑ f α g α (ˆr), l = m=−l
α=
()
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
whereby the Slepian-basis expansion coefficients are obtained as f α = ∫ f (ˆr)g α (ˆr) dΩ.
()
Ω
If the function of interest is spatially localized in the region R, a truncated reconstruction using Slepian functions built for the same region will constitute a very good, and sparse, local approximation to it (Simons et al. ): N D
f (ˆr) ≈ ∑ f α g α (ˆr),
ˆr ∈ R.
()
α=
We represent any sampled, bandlimited function f by an M-dimensional column vector f = ( f ⋯ f j ⋯ f M )T ,
()
where f j = f (ˆr j ) is the value of f at pixel j, and M is the total number of sampling locations. In the most general case, the distribution of pixel centers will be completely arbitrary. The special case of equal-area pixelization of a -D function f (ˆr) on the unit sphere Ω is analogous to the equispaced digitization of a -D time series. Integrals will then be assumed to be approximated with sufficient accuracy by a Riemann sum over a dense set of pixels, M
∫ f (ˆr) dΩ ≈ ΔΩ ∑ f j
and
j=
T ∫ f (ˆr) dΩ ≈ ΔΩ f f.
()
We have deliberately left the integration domain out of the above equations to cover both sampling over the entire unit sphere surface Ω, in which case the solid angle ΔΩ = π/M (case ) as well as over an incomplete subdomain R ⊂ Ω, in which case ΔΩ = A/M, with A the area of the region R (case ). If we collect the real spherical harmonic basis functions Yl m into an (L + ) × M-dimensional matrix ⎛ Y (ˆr ) ⎜ ⎜ Y=⎜ ⋯ ⎜ ⎜ ⎜ ⎝ YLL (ˆr )
⋯
⋯
Y (ˆr j ) ⋮ Yl m (ˆr j ) ⋮ YLL (ˆr j )
⋯
Y (ˆr M ) ⎞ ⎟ ⎟ ⎟, ⋯ ⎟ ⎟ ⎟ YLL (ˆr M ) ⎠
⋯
()
and the spherical harmonic coefficients of the function into an (L + ) × -dimensional vector T
f = ( f ⋯ f l m ⋯ f LL ) ,
()
we can write the spherical harmonic synthesis in Eq. for sampled data without loss of generality as T () f = Y f. We will adhere to the notation convention of using sans-serif fonts (e.g., f, Y) for vectors or matrices that depend on at least one spatial variable, and serifed fonts (e.g., f, D ) for those that are entirely composed of “spectral” quantities. In the case of dense, equal-area, whole-sphere sampling we have an approximation to Eq. : YYT ≈ ΔΩ− I
(case ),
()
where the elements of the (L + ) × (L + ) -dimensional spectral identity matrix I are given by the Kronecker deltas δ l l ′ δ mm′ . In the case of dense, equal-area, sampling over some closed
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
region R, we find instead an approximation to the (L+) ×(L+)-dimensional “spatiospectral localization matrix”: YYT ≈ ΔΩ− D
(case ),
()
where the elements of D are those defined in Eq. b. Let us now introduce the (L + ) × (L + ) -dimensional matrix of spectral Slepian eigenfunctions by ⎛ g ⎜ ⎜ G=⎜ ⎜ ⋯ ⎜ ⎜ ⎝ g LL
⋯
g α ⋮ gl m α ⋮ g LL α
⋯
⋯
⋯
g (L+) ⎞ ⎟ ⎟ ⎟. ⋯ ⎟ ⎟ ⎟ g LL (L+) ⎠
()
This is the matrix that contains the eigenfunctions of the problem defined in Eq. , which we rewrite as D G = G Λ,
()
where the diagonal matrix with the concentration eigenvalues is given by Λ = diag ( λ ⋯ λ α ⋯ λ(L+) ) .
()
The spectral orthogonality relations of Eq. are GT G = I,
GT D G = Λ,
()
where the elements of the (L + ) × (L + ) -dimensional Slepian identity matrix I are given by the Kronecker deltas δ α β . We write the Slepian functions of Eq. as G = GT Y
and Y = GG,
()
where the (L + ) × M-dimensional matrix holding the sampled spatial Slepian functions is given by g (ˆr ) ⎛ ⎜ ⎜ G=⎜ ⋯ ⎜ ⎜ ⎜ ⎝ g(L+) (ˆr )
⋯
⋯
g (ˆr j ) ⋮ g α (ˆr j ) ⋮ g(L+) (ˆr j )
⋯
⋯
g (ˆr M )
⎞ ⎟ ⎟ ⎟. ⋯ ⎟ ⎟ ⎟ g(L+) (ˆr M ) ⎠
()
Under a dense, equal-area, whole-sphere sampling, we will recover the spatial orthogonality of Eq. approximately as T
−
GG ≈ ΔΩ I (case ),
()
whereas for dense, equal-area, sampling over a region R we will get, instead, T
−
GG ≈ ΔΩ Λ
(case ).
()
With this matrix notation we shall revisit both estimation problems of the previous section.
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
.
Problem (i), Revisited
..
Spherical Harmonic Solution
If we treat Eq. as a noiseless inverse problem in which the sampled data f are given but from which the coefficients f are to be determined, we find that for dense, equal-area, whole-sphere sampling, the solution ˆf ≈ ΔΩ Y f (case ) () is simply the discrete approximation to the spherical harmonic analysis formula (). For dense, equal-area, regional sampling we need to calculate fˆ ≈ ΔΩ D− Yf
(case ).
()
Both of these cases are simply the relevant solutions to the familiar overdetermined spherical harmonic inversion problem (Kaula ; Menke ; Aster et al. ) for discretely sampled data, i.e., the least-squares solution to Eq. , ˆf = (YYT )− Y f,
()
for the particular cases described by Eqs. –. In Eq. , we furthermore recognize the discrete version of Eq. with η = , the undamped solution to the minimum mean-squared error inverse problem posed in continuous form in Eq. . From the continuous limiting case, Eq. , we thus discover the general form that damping should take in regularizing the ill-conditioned inverse required in Eqs. –. Its principal property is that it differs from the customary ad hoc practice of adding small values on the diagonal only. Finally, in the most general and admittedly most commonly encountered case of randomly scattered data we require the Moore–Penrose pseudo-inverse ˆf = pinv(YT )f, () which is constructed by inverting the singular value decomposition (svd) of YT with its singular values truncated beyond where they fall below a certain threshold (Xu ). Solving Eq. by truncated svd is equivalent to inverting a truncated eigenvalue expansion of the normal matrix YYT as it appears in Eq. , as can be easily shown.
..
Slepian Basis Solution
If we collect the Slepian expansion coefficients of the function f into the (L+) ×-dimensional vector T t = ( f ⋯ f α ⋯ f(L+) ) , () the expansion () in the Slepian basis takes the form T
T
f = G t = Y Gt,
()
where we used Eqs. – to obtain the second equality. Comparing Eq. with Eq. , we see that the Slepian expansion coefficients of a function transform to and from the spherical harmonic coefficients as: T f = Gt and t = G f. ()
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
Under dense, equal-area, sampling with complete coverage the coefficients in Eq. can be estimated from ˆt ≈ ΔΩ Gf (case ), () the discrete, approximate, version of Eq. . For dense, equal-area, sampling in a limited region R we get ˆt ≈ ΔΩ Λ− G f (case ). () As expected, both of the solutions (–) are again special cases of the overdetermined leastsquares solution ˆt = (GGT )− Gf, () as applied to Eqs. –. We encountered Eq. before in the continuous form of Eq. ; it solves the undamped minimum mean-squared error problem (). The regularization of this illconditioned inverse problem may be achieved by truncation of the concentration eigenvalues, e.g., by restricting the size of the (L + ) × (L + ) -dimensional operator GGT to its first J × J subblock. Finally, in the most general, scattered-data case, we would be using an eigenvaluetruncated version of Eq. , or, which is equivalent, form the pseudo-inverse ˆt = pinv(GT )f.
()
The solutions (–) and (–) are equivalent and differ only by the orthonormal change of basis from the spherical harmonics to the Slepian functions. Indeed, using Eqs. and to transform Eq. into an equation for the spherical harmonic coefficients, and comparing with Eq. exposes the relation G(GGT )− GT = (YYT )− ,
()
which is a trivial identity for case (insert Eqs. , , and ) and, after substituting Eqs. and , entails GΛ− GT = D− () for case , which holds by virtue of Eq. . Equation can also be verified directly from Eq. , which implies T T T () YY = G(GG )G . The popular but labor-intensive procedure by which the unknown spherical harmonic expansion coefficients of a scattered data set are obtained by forming the Moore–Penrose pseudoinverse as in Eq. is thus equivalent to determining the truncated Slepian solution of Eq. in the limit of continuous and equal-area, but incomplete data coverage. In that limit, the generic eigenvalue decomposition of the normal matrix becomes a specific statement of the Slepian problem as we encountered it before, namely YY TΔΩ = UΣ UT
→
D = GΛGT .
()
Such a connection has been previously pointed out for time series (Wingham ) and leads to the notion of “generalized prolate spheroidal functions” (Bronez ) should the “Slepian” functions be computed from a formulation of the concentration problem in the scattered data space directly, rather than being determined by sampling those obtained from solving the corresponding continuous problem, as we have described here. Above, we showed how to stabilize the inverse problem of Eq. by damping. We dealt with the case of continuously available data only; the form in which it appears in Eq. makes
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
it clear that damping is hardly practical for scattered data. Indeed, it requires knowledge of the ¯ in addition to being sensitive to the choice of η, whose complementary localization operator D, optimal value depends implicitly on the unknown signal-to-noise ratio (Simons and Dahlen ). The data-driven approach taken in Eq. is the more sensible one (Xu ). We have now seen that, in the limit of continuous partial coverage, this corresponds to the optimal solution of the problem formulated directly in the Slepian basis. It is consequently advantageous to also work in the Slepian basis in case the data collected are scattered but closely collocated in some region of interest. Prior knowledge of the geometry of this region and a prior idea of the spherical harmonic bandwidth of the data to be inverted allows us to construct a Slepian basis for the situation at hand, and the problem of finding the Slepian expansion coefficients of the unknown underlying function can be solved using Eqs. –. The measure within which this approach agrees with the theoretical form of Eq. will depend on how regularly the data are distributed within the region of study, i.e., on the error in the approximation (). But if indeed the scattered-data Slepian normal matrix GGT is nearly diagonal in its first J × J-dimensional block due to the collocated observations having been favorably, if irregularly, distributed, then Eq. , which, strictly speaking, requires no matrix inversion, can be applied directly. If this is not the case, but the data are still collocated or we are only interested in a local approximation to the unknown signal, we can restrict G to its first J rows, prior to diagonalizing GGT or performing the svd of a partial GT as necessary to calculate Eqs. –. Compared to solving Eqs. –, the computational savings will still be substantial, as only when R ≈ Ω will the operator YYT be nearly diagonal. Truncation of the eigenvalues of YYT is akin to truncating the matrix GGT itself, which is diagonal or will be nearly so. With the theoretically, continuously determined, sampled Slepian functions as a parametrization, the truncated expansion is easy to obtain and the solution will be locally faithful within the region of interest R. In contrast, should we truncate YYT itself, without first diagonalizing it, we would be estimating a low-degree approximation of the signal which would have poor resolution everywhere.
..
Bias and Variance
For completeness we briefly return to the expressions for the mean-squared estimation error of the damped spherical-harmonic and the truncated Slepian function methods, Eqs. –, which we quoted for the example of “white” signal and noise. Introducing the (L + ) × (L + ) -dimensional spectral matrices H = Λ + η (I − Λ), V = NH− Λ,
and B =
√
S H− (I − Λ),
(a) (b)
we handily rewrite the “full” version of Eq. in two spatial variables as the error covariance matrix ⟨є(ˆr)є(ˆr′ )⟩ = GT(V + η B )G.
()
We subdivide the matrix with Slepian functions into the truncated set of the best-concentrated α = → J and the complementary set of remaining α = J + → (L + ) functions, as follows T
G=(G ¯
¯ T )T , G
()
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
and similarly separate the eigenvalues, writing ¯ = diag ( λ ⋯ λ J ) , Λ
(a)
Λ = diag ( λ J+ ⋯ λ(L+) ) . (b) ¯ Likewise, the identity matrix is split into two parts, ¯I and I. If we now also redefine ¯ √ ¯ = S ¯I, ¯ = NΛ ¯ − , and B (a) V √ − (b) V = N Λ , and B = S I, ¯ ¯ ¯ ¯ the equivalent version of Eq. is readily transformed into the full spatial error covariance matrix ¯ TB ¯ ¯ G. ⟨є(ˆr)є(ˆr′ )⟩ = GT V G + G () ¯ ¯ ¯ In selecting the Slepian basis we have thus successfully separated the effect of the variance and the bias on the mean-squared reconstruction error of a noisily observed signal. If the region of observation is a contiguous closed domain R ⊂ Ω and the truncation should take place at the Shannon number N D , we have thereby identified the variance as due to noise in the region where data are available, and the bias to signal neglected in the truncated expansion— which, in the proper Slepian basis, corresponds to the regions over which no observations exist. In practice, the truncation will happen at some J which depends on the signal-to-noise ratio (Simons and Dahlen ). Finally, we shall also apply the notions of discretely acquired data to the solutions of problem (ii), below.
.
Problem (ii), Revisited
We need two more pieces of notation in order to rewrite the expressions for the spectral estimates () and () in the “pixel-basis.” First we construct the M × M-dimensional symmetric spatial matrix collecting the fixed-degree Legendre polynomials evaluated at the angular distances between all pairs of observations points, ⎛ Pl (ˆr ⋅ ˆr ) ⎜ l + ⎜ Pl = ( ⋯ )⎜ π ⎜ ⎜ ⎜ ⎝ Pl (ˆr M ⋅ ˆr )
⋯
⋯
Pl (ˆr ⋅ ˆr j ) ⋮ Pl (ˆr i ⋅ rˆ j ) ⋮ Pl (ˆr M ⋅ ˆr j )
⋯
⋯
Pl (ˆr ⋅ ˆr M ) ⎞ ⎟ ⎟ ⎟. ⋯ ⎟ ⎟ ⎟ Pl (ˆr M ⋅ ˆr M ) ⎠
()
l Yl m (ˆr i )Yl m (ˆr j ), by the addition theorem, Eq. . And The elements of P l are thus ∑m=−l α finally, we define G l , the M × M symmetric matrix with elements given by
(Gαl ) i j = (
..
l + ) g α (ˆr i )Pl (ˆr i ⋅ ˆr j )g α (ˆr j ). π
()
The Spherical Periodogram
The expression equivalent to Eq. is now written as π (ΔΩ) T SˆSP ) d P l d, l =( A l +
()
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
whereby the column vector d contains the sampled data as in the notation for Eq. . This lends itself easily to computation, and the statistics of Eqs. – hold, approximately, for sufficiently densely sampled data.
..
The Spherical Multitaper Estimate
Finally, the expression equivalent to Eq. becomes (L+) π (ΔΩ) T α = ∑ λ α ( D ) SˆMT d Gl d. l N l + α=
()
Both Eqs. and are quadratic forms, earning them the nickname “quadratic spectral estimators” (Mullis and Scharf ). The key difference with the maximum-likelihood estimator popular in cosmology (Bond et al. ; Oh et al. ; Hinshaw et al. ), which can also be written as a quadratic form (Tegmark ), is that neither P l nor Gαl depend on the unknown spectrum itself. In contrast, maximum-likelihood estimation is inherently non linear, requiring iteration to converge to the most probable estimate of the power spectral density (Dahlen and Simons ). The estimate () has the statistical properties that we listed earlier as Eqs. –. These continue to hold when the data pixelization is fine enough to have integral expressions of the kind () be exact. As mentioned before, for completely irregularly and potentially non-densely distributed discrete data on the sphere, “generalized” Slepian functions (Bronez ) could be constructed specifically for the purpose of their power spectral estimation, and used to build the operator ().
Conclusions
What is the information contained in a bandlimited set of scientific observations made over an incomplete, e.g., temporally or spatially limited sampling domain? How can this “information,” e.g., an estimate of the signal itself, or of its energy density, be determined from noisy data, and how shall it be represented? These seemingly age-old fundamental questions, which have implications beyond the scientific (Slepian ), had been solved—some say, by conveniently ignoring them—heuristically, by engineers, well before receiving their first satisfactory answers given in the theoretical treatment by Slepian, Landau, and Pollak (Slepian and Pollak ; Landau and Pollak , ); first for “continuous” time series, later generalized to the multidimensional and discrete cases (Slepian , ; Bronez ). By the “Slepian functions” in the title of this contribution, we have lumped together all functions that are “spatiospectrally” concentrated, quadratically, in the original sense of Slepian. In one dimension, these are the “prolate spheroidal functions” whose popularity is as enduring as their utility. In two Cartesian dimensions, and on the surface of the unit sphere, their time is yet to come, and we have made a case for it in this study. The answers to the questions posed above are as ever relevant for the geosciences of today. There, we often face the additional complications of irregularly shaped study domains, scattered observations of noise-contaminated potential fields, perhaps collected from an altitude
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
above the source by airplanes or satellites, and an acquisition and model-space geometry that is rarely if ever non-spherical. Thus the Slepian functions are especially suited for geoscientific applications and to study any type of geographical information, in general. Two problems that are of particular interest in the geosciences, but also further afield, are how to form a statistically “optimal” estimate of the signal giving rise to the data, and how to estimate the power spectral density of such signal. The first, an inverse problem whose solution is linear in the data, applies to forming mass flux estimates from time-variable gravity, e.g., by the grace mission, or to the characterization of the terrestrial magnetic field by satellites such as champ or swarm. The second, which is quadratic in the data, is of interest in studying the statistics of the Earth’s or planetary topography, and especially for the cross-spectral analysis of gravity and topography, which can yield important clues about the internal structure of the planets. The second problem is also of great interest in cosmology, where missions such as wmap and planck are mapping the cosmic microwave background radiation, which is best modeled spectrally to constrain models of the evolution of our universe. Slepian functions, as we have shown by focusing on the case of spherical geometry, provide the mathematical framework to solve such problems. They are a convenient and easily obtained doubly-orthogonal mathematical basis in which to express, and thus by which to recover, signals that are geographically localized, or incompletely (and noisily) observed. For this they are much better suited than the traditional Fourier or spherical harmonic bases, and they are more “geologically intuitive” than wavelet bases in retaining a firm geographic footprint and preserving the traditional notions of frequency or spherical harmonic degree. They are furthermore extremely performant as data tapers to regularize the inverse problem of power spectral density determination from noisy and patchy observations, which can then be solved satisfactorily without costly iteration. Finally, by the interpretation of the Slepian functions as their limiting cases, much can be learned about the statistical nature of such inverse problems when the data provided are themselves scattered within a specific areal region of study.
Acknowledgments For this work, I am most indebted to Tony Dahlen (–), for sharing my interest in this subject and being my friend, teacher, and mentor until his untimely death. I also thank Mark Wieczorek and Volker Michel for many enlightening discussions on this and similar topics over the last several years. Dong V. Wang aided with the computations for the Cartesian case. Financial support for this work was provided by the U. S. National Science Foundation under Grants EAR- and EAR-, and by the Université Paris Diderot–Paris and the Institut de Physique du Globe de Paris in St. Maur-des-Fossés, where this contribution was completed. Computer algorithms are made available on www.frederik.net.
References Albertella A, Sansò F, Sneeuw N () Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere. J Geodesy :–
Amirbekyan A, Michel V, Simons FJ () Parameterizing surface-wave tomopgraphic models with harmonic spherical splines. Geophys J Int ():, doi: ./j.–X...x
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
Aster RC, Borchers B, Thurber CH () Parameter estimation and inverse problems, vol of International Geophysics Series. Academic Press, San Diego Bendat JS, Piersol AG () Random data: Analysis and measurement procedures, rd edn. Wiley, New York Bond JR, Jaffe AH, Knox L () Estimating the power spectrum of the cosmic microwave background. Phys Rev D ():– Bronez TP () Spectral estimation of irregularly sampled multidimensional processes by generalized prolate spheroidal sequences. IEEE Trans Acoust Speech Signal Process ():– Chave AD, Thomson DJ, Ander ME () On the robust estimation of power spectra, coherences, and transfer functions. J Geophys Res (B):– Cohen L () Time-frequency distributions—a review. Proc IEEE ():– Cox DR, Hinkley DV () Theoretical statistics. Chapman and Hall, London Dahlen FA, Tromp J () Theoretical global seismology. Princeton Univ. Press, Princeton Dahlen FA, Simons FJ () Spectral estimation on a sphere in geophysics and cosmology. Geophys J Int :–, doi: ./j.– X...x Daubechies I () Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory :– Daubechies I () The wavelet transform, timefrequency localization and signal analysis. IEEE Trans Inform Theory :– Daubechies I () Ten lectures on wavelets, vol of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial & Applied Mathematics, Philadelphia Daubechies I, Paul T () Time-frequency localisation operators—a geometric phase space approach: II. The use of dilations. Inv Probl ():– Donoho DL, Stark PB () Uncertainty principles and signal recovery. SIAM J Appl Math ():– Edmonds AR () Angular momentum in quantum mechanics. Princeton Univ. Press, Princeton Flandrin P () Temps-Fréquence, nd edn. Hermès, Paris Freeden W, Windheuser U () Combined spherical harmonic and wavelet expansion—a future concept in Earth’s gravitational determination. Appl Comput Harmon Anal :–
Freeden W, Gervens T, Schreiner M () Constructive approximation on the sphere. Clarendon Press, Oxford Gilbert EN, Slepian D () Doubly orthogonal concentrated polynomials. SIAM J Math Anal ():– Grishchuk LP, Martin J () Best unbiased estimates for the microwave background anisotropies. Phys Rev D ():– Grünbaum FA () Eigenvectors of a Toeplitz matrix: discrete version of the prolate spheroidal wave functions. SIAM J Alg Disc Meth ():– Han S-C, Simons FJ () Spatiospectral localization of global geopotential fields from the Gravity Recovery and Climate Experiment (GRACE) reveals the coseismic gravity change owing to the Sumatra-Andaman earthquake. J Geophys Res :B, doi: ./JB Hanssen A () Multidimensional multitaper spectral estimation. Signal Process :– Hauser MG, Peebles PJE () Statistical analysis of catalogs of extragalactic objects. II. The Abell catalog of rich clusters. Astrophys. J :– Hinshaw G, Spergel DN, Verde L, Hill RS, Meyer SS, Barnes C, Bennett CL, Halpern M, Jarosik N, Kogut A, Komatsu E, Limon M, Page L, Tucker GS, Weiland JL, Wollack E, Wright EL () First-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: the angular power spectrum. Astrophys. J Supp Ser :– Hivon E, Górski KM, Netterfield CB, Crill BP, Prunet S, Hansen F () MASTER of the cosmic microwave background anisotropy power spectrum: a fast method for statistical analysis of large and complex cosmic microwave background data sets. Astrophys. J :– Jones RH () Stochastic processes on a sphere. Ann Math Stat ():– Kaula WM () Theory of statistical analysis of data distributed over a sphere. Rev Geophys ():– Knox L () Determination of inflationary observables by cosmic microwave background anisotropy experiments. Phys Rev D ():– Landau HJ () On the eigenvalue behavior of certain convolution equations. Trans Am Math Soc :– Landau HJ, Pollak HO () Prolate spheroidal wave functions, Fourier analysis and uncertainty—II. Bell Syst Tech J ():– Landau HJ, Pollak HO () Prolate spheroidal wave functions, Fourier analysis and uncertainty— III: the dimension of the space of essentially
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
time- and band-limited signals. Bell Syst Tech J ():– Mallat S () A wavelet tour of signal processing. Academic, San Diego Menke W () Geophysical data analysis: discrete inverse theory, vol. of International Geophysics Series, nd Rev. edn. Academic Press, San Diego Messiah A () Quantum mechanics. Dover, New York Mullis CT, Scharf LL () Quadratic estimators of the power spectrum. In: Haykin S (ed) Advances in Spectrum analysis and array processing, vol , chap. . Prentice-Hall, Englewood Cliffs, pp – Nashed MZ, Walter GG () General sampling theorems for functions in reproducing kernel Hilbert spaces. Math Control Signals Syst : – Oh SP, Spergel DN, Hinshaw G () An efficient technique to determine the power spectrum from cosmic microwave background sky maps. Astrophys. J :– Peebles PJE () Statistical analysis of catalogs of extragalactic objects. I. Theory. Astrophys. J :– Percival DB, Walden AT () Spectral analysis for physical applications, multitaper and conventional univariate techniques. Cambridge Univ. Press, New York Riedel KS, Sidorenko A () Minimum bias multiple taper spectral estimation. IEEE Trans Signal Process ():– Schuster A () An investigation of hidden periodicities with application to a supposed -day period of meteorological phenomena. Terr Magn :– Shkolnisky Y () Prolate spheroidal wave functions on a disc—integration and approximation of two-dimensional bandlimited functions. Appl Comput Harmon Anal :–, doi: ./j.acha... Simons FJ, Dahlen FA () Spherical Slepian functions and the polar gap in geodesy. Geophys J Int :–, doi: ./j.– X...x Simons FJ, Dahlen FA () A spatiospectral localization approach to estimating potential fields on the surface of a sphere from noisy, incomplete data taken at satellite altitudes. In: Van de Ville D, Goyal VK, Papadakis M (eds) Wavelets, vol , p , doi: ./ ., Proc. SPIE Simons FJ, Wang DV () Spatiospectral concentration in the Cartesian plane, manuscript in preparation
Simons FJ, Zuber MT, Korenaga J () Isostatic response of the Australian lithosphere: estimation of effective elastic thickness and anisotropy using multitaper spectral analysis. J Geophys Res :– Simons FJ, Dahlen FA, Wieczorek MA () Spatiospectral concentration on a sphere. SIAM Rev ():–, doi: ./S Simons FJ, Hawthorne JC, Beggan CD () Efficient analysis and representation of geophysical processes using localized spherical basis functions. In Van de Ville D, Goyal VK, Papadakis M (eds), Wavelets XIII, vol. (G), doi: ./., Proc. SPIE Slepian D, Pollak HO () Prolate spheroidal wave functions, Fourier analysis and uncertainty—I. Bell Syst Tech J ():– Slepian D () Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst Tech J ():– Slepian D, Sonnenblick E () Eigenvalues associated with prolate spheroidal wave functions of zero order. Bell Syst Tech J ():– Slepian D () On bandwidth. Proc IEEE ():– Slepian D () Prolate spheroidal wave functions, Fourier analysis and uncertainty—V: the discrete case. Bell Syst Tech J :– Slepian D () Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev (): – Sneeuw N, van Gelderen M () The polar gap. In: Sansò F, Rummel R (eds) Geodetic boundary value problems in view of the one centimeter geoid, Lecture notes in Earth sciences . Springer, Berlin, pp – Tegmark M () How to measure CMB power spectra without losing information. Phys Rev D ():– Tegmark M, Taylor AN, Heavens AF () Karhunen-Loève eigenvalue problems in cosmology: how should we tackle large data sets? Astrophys. J ():– Thomson DJ () Spectrum estimation and harmonic analysis. Proc IEEE :– Thomson DJ () Jackknifing multitaper spectrum estimates. IEEE Signal Process Mag :–, doi: ./MSP.. Thomson DJ, Chave AD () Jackknifed error estimates for spectra, coherences, and transfer functions. Haykin S (ed) Advances in spectrum analysis and array processing, vol , chap. . Prentice-Hall, Englewood Cliffs, pp –
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
Varshalovich DA, Moskalev AN, Kherso n´ skii VK () Quantum theory of angular momentum. World Scientific, Singapore de Villiers GD, Marchaud FBT, Pike ER () Generalized Gaussian quadrature applied to an inverse problem in antenna theory: II. The two-dimensional case with circular symmetry. Inverse Probl :– Wieczorek MA, Simons FJ () Localized spectral analysis on the sphere. Geophys J Int ():– , doi: ./j.–X...x Wieczorek MA, Simons FJ () Minimumvariance spectral analysis on the sphere. J Fourier Anal Appl (): –, doi: ./ s---
Wingham DJ () The reconstruction of a bandlimited function and its Fourier transform from a finite number of samples at arbitrary locations by singular value decomposition. IEEE Trans Signal Process ():–, doi: ./ . Xu P () Determination of surface gravity anomalies using gradiometric observables. Geophys J Int :– Xu P () Truncated SVD methods for discrete linear ill-posed problems. Geophys J Int ():– Yao K () Application of reproducing kernel Hilbert spaces—bandlimited signal models. Inform Control ():–
